The Jamming Transition in Granular Systems

T. S. Majmudar1, M. Sperl1, S. Luding2, R.P. Behringer11Duke University, Department of Physics, Box 90305, Durham,NC 27708, USA,

2Technische Universiteit Delft, DelftChemTech, Particle Technology,Nanostructured Materials, Julianlaan 136, 2628 BL Delft, The Netherlands

(Dated: February 2, 2007)

Recent simulations have predicted that near jamming for collections of spherical particles, there will be adiscontinuous increase in the mean contact number,Z, at a critical volume fraction,φc. Above φc, Z and thepressure,P, are predicted to increase as power laws inφ − φc. In experiments using photoelastic disks wecorroborate a rapid increase inZ at φc and power-law behavior aboveφc for Z and P. Specifically we findpower-law increase as a function ofφ−φc for Z−Zc with an exponentβ around 0.5, and forP with an exponentψ around 1.1. These exponents are in good agreement with simulations. We also find reasonable agreementwith a recent mean-field theory for frictionless particles.

PACS numbers: 64.60.-i,83.80.Fg,45.70.-n

A solid, in contrast to a fluid, is characterized by me-chanical stability that implies a finite resistance to shearandisotropic deformation. While such stability can originatefromlong-range crystalline order, there is no general agreement onhow mechanical stability arises for disordered systems, suchas molecular and colloidal glasses, gels, foams, and granularpackings [1]. For a granular system in particular, a key ques-tion concerns how stability occurs when the packing fraction,φ, increases from below to above a critical valueφc for whichthere are just enough contacts per particle,Z, to satisfy theconditions of mechanical stability. In recent simulationsonfrictionless systems it was found thatZ exhibits a disconti-nuity at φc followed by a power law increase forφ > φc [2–5]. The pressure is also predicted to increase as a power-lawaboveφc.

A number of recent theoretical studies address jamming,and we note work that may be relevant to granular systems.Silbert, O’Hern et al. have shown in computer simulationsof frictionless particles [2–4] that: a) for increasingφ, Zincreases discontinuously at the transition point from zeroto a finite number,Zc, corresponding to the isostatic value(needed for mechanical stability); b) for both two- and three-dimensional systems,Z is expected to continue increasing as(φ− φc)

β aboveφc, whereβ = 0.5; c) the pressure,P, is ex-pected to grow aboveφc as(φ− φc)

ψ, whereψ = α f − 1 inthe simulations, andα f is the exponent for the interparticlepotential. More recent simulations by Donev et. al. for hardspheres in three dimensions found a slightly higher value forβ, β ≈ 0.6, in maximally random jammed packings [5]. It isinteresting to note that a model for foam exhibits quite similarbehavior forZ [6]. Henkes and Chakraborty [7] constructeda mean field theory of the jamming transition in 2D based onentropy arguments. These authors predict power-law scalingfor P andZ in terms of a variableα, which is the pressurederivative of the entropy. By eliminatingα, one obtains an al-gebraic relation betweenP andZ−Zc from these predictions,which we present below in the context of our data.

While the simulations agree among themselves at leastqualitatively, so far, these novel features have not been iden-

tified in experiments. Hence, it is crucial to test these pre-dictions experimentally. In the following, we present experi-mental data forZ andP vs. φ, based on a method that yieldsaccurate determination of the of contacts and identifies powerlaws inZ andP for a two-dimensional experimental system ofphotoelastic disks. By measuring bothP andZ, we can alsoobtain a sharper value for the critical packing fractionφc, forthe onset of jamming, and we can test the model of Henkesand Chakraborty.

The relevant simulations have been carried out predomi-nantly for frictionless particles. For real frictional particlesthere will clearly be some differences. For instance, in theisostatic limit,Z equals 4 for frictionless disks, whereas forfrictional disks,Z is around 3, depending on the system de-tails [8]. Other predictions such as specific critical exponentsmay also need modification. However, one might hope thatthe observed experimental behavior, in particular critical ex-ponents, might be similar to that for frictionless particles ifthe frictional forces are typically small relative to the normalforces. Indeed, in recent experiments, the typical inter-grainfrictional forces in a physical granular system were found tobe only about 10% of the normal forces [9].

Fig. 1a shows a schematic of the apparatus. We use abidisperse mixture (80% small and 20% large particles) ofapproximately 3000 polymer (PSM-4) photoelastic (birefrin-gent under stress) disks with diameter 0.74 cm or 0.86 cm.This ratio preserves a disordered system. The disks haveYoung’s modulus of 4 MPa, and a static coefficient of frictionof 0.85. The model granular system is confined in a biaxialtest cell (42cm×42cm with two movable walls) which restson a smooth Plexiglas sheet. The displacements of the wallscan be set very precisely with stepper motors. The linear dis-placement step size used in this experiment is 40µm, whichis approximately 0.005D, whereD is the average diameter ofthe disks. The deformationδ per particle is less than 1% in thecompressed state. The setup is horizontal and placed betweencrossed circular polarizers. It is imaged from above with an8MP CCD color camera which captures roughly 1200 disks inthe center of the cell, enabling us to visualize the stress field

2

FIG. 1: (a) Schematic cross-section of biaxial cell experiment (notto scale). Two walls can be moved independently to obtain a de-sired sample deformation. (b) Examples of contacts and particlesthat are either close but not actually in contact, or contacts with verysmall forces. Circles show true contacts, squares show false apparentcontacts. (c) Image of a single disk at the typical resolution of theexperiment. (d) Sample image of highly jammed/compressed stateand (e) almost unjammed state.

within each disk (Fig. 1). We then obtain good measurementsof the vector contact forces (normal and tangential = frictionalcomponents) [9].

We also use the particle photoelasticity to accurately deter-mine the presence or absence of contacts between particles.In numerical studies one can use a simple overlap criterionto determine contacts: a contact occurs if the distance be-tween particle centers is smaller than the sum of the particleradii. However, in experimental systems, a criterion basedsolely on the particle centers is susceptible to relativelylargeerrors which include false positives (Fig. 1b - squares) as wellas false negatives (circles). As seen in Fig. 1b, the contactsthrough which there is force transmission appear as sourcepoints for the stress pattern. Further details are given in thesupplementary material, section I.

We use two protocols to produce different packing frac-tions: we either compress the system from an initially stress-free state, or decompress the system until the end state is es-sentially a stress-free state. The results for both protocols arethe same within error bars aboveφc; below jamming, the datafor Z obtained by compression are a few percent below thosefor decompression. Below, we will present decompressiondata. Figures 1d,e show the initial highly stressed state andthe end state after decompression, respectively. After eachdecompression step, we apply tapping to relax stress in thesystem. This could be seen as roughly analogous to the an-nealing process invoked in some simulations. Two images arecaptured at each state: one without polarizers to determinethedisk centers and one with polarizers to record the stress.

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FIG. 2: Average contact number and pressure at the jamming tran-sition. Top and bottom panels showZ−Zc andP vs φ−φc, respec-tively, with rattlers included (stars) or excluded (diamonds). Dashedand full curves in the top panel give power-law fits(φ− φc)

β withβ = 0.495 and 0.561 for the case with and without rattlers, respec-tively. Full curve in the lower panel gives the fit(φ − φc)

ψ withψ = 1.1; dashed line shows a linear law for comparison. Inset:Zvs φ for a larger range inφ.

The averageZ can be computed either by counting only theforce bearing disks or by counting all the disks including rat-tlers which do not contribute to the mechanical stability ofthesystem. We consider as rattlers, all the disks which have lessthan 2 contacts. For the number of rattlers beyond the transi-tion point we find an exponential decrease withφ−φc; hence,a divergence in the number of rattlers atφc is not indicated bythe data.

We next compute the Cauchy stress tensor for each disk,σi j = 1

2A ∑(Fix j +Fjxi); P, is the trace of this tensor. Here,A is the Voronoi area for the given disk, and the sum is takenover contacts for a given disk. We then compute the average ofthe pressure over the ensemble of disks in the system. For thedata presented below, we performed two sets of experiments:one with a larger range, 0.8390≤ φ ≤ 0.8650, and also largerstep size,∆φ = 0.016, and – after the jamming region wasidentified – a second set at a finer scale with 0.840745≤ φ ≤

0.853312, with a step size,∆φ = 0.000324.

3

The inset in Fig. 2 shows data forZ over a broad range ofφ (with rattlers–stars; without–squares). These data show asignificant rise inZ at the jamming transition. While this riseis not sharply discontinuous, it occurs over a very small rangein φ. At higherφ, the variations of the curves are similar withand without rattlers. At lowerφ, their behavior differs: Thevalues of Z drop lower for the case with rattlers. The pressureP(φ− φc) in Fig. 2 shows a flat background below jamming,and then a sharp positive change in slope at a well definedφ.The pressure is not identically zero below jamming for similarreasons that the jump inZ is not perfectly sharp, as discussedbelow.

To compare these experimental results to predictions aboveφc, we carry out least squares fits ofZ−Zc andP to φ− φc.These fits depend on the choice ofφc, which has some ambi-guity due to the rounding; the data allow a range from around0.840 to 0.843. In fact,φc can be determined in several ways:the point whereZ reaches 3, the point whereP begins to riseabove the background, etc. (cf. supplemetary material). Weshow results of these fits in Fig. 2, starting with the upperpanel, which shows power-law fits(Z−Zc) ∝ (φ−φc)

β. Thefitted exponentβ depends on the choice ofφc but the variationis small without rattlers, 0.494≤ β ≤ 0.564, and somewhatlarger with rattlers, 0.363≤ β ≤ 0.525. The details for sev-eral different specific fits are given in the supplementary ma-terial, section II. The pointφc = 0.84220 whereP rises abovethe background level is used in Fig. 2, and yields a consis-tent fit for bothP andZ. The point whereZ reaches 3 for thecase without rattlers agrees with the previous case to withinδφc = 0.0005, and the exponents are quite similar. Comparingwith the simulations for frictionless particles, we find that ourvalues ofβ ≈ 0.55 for the data without rattlers are larger thanthe value of 0.5 reported in [2, 3], but smaller than those ofDonev et. al. [5] obtaining 0.6 in 3D. In contrast, for a modelof frictional disks under shear, Aharonov and Sparks [10] ob-tain the much lower value of 0.36. However, a direct com-parison is not possible to the present case of jamming underisotropic conditions.

Figure 2 shows the variation ofP with φ in the lower panel,indicating a clear transition atφc = 0.8422±0.0005. For thischoice ofφc, P increases asP ∝ (φ−φc)

ψ with ψ = 1.1±0.05aboveφc. This value ofψ pertains to a fit over the full rangeφ ≥ φc of Fig. 2; a larger exponent would be obtained if the fitrange were limited to very close toφc. This value is close tothe valueψ = 1.0 found [2, 3] for a linear force law, and thislinear law is indicated as a dashed line in Fig. 2. One expectssuch a linear force law (with a logarithmic correction) for idealdisks, but direct mechanical calibration of the force law for thecylinders is closer toδ3/2 (see supplementary material). Thisrather high exponent for the force law is attributable to thesmall asperities, which influence the force law for small de-formations. However, the photoelastic response is detectableonly for δ > 150µm, and for suchδ’s, the force law is close tolocally linear inδ.

From theP vs. φ data, we can also obtain the bulk modulus,B = −A∂P/∂A, whereA is the area enclosed by the system

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0.82 0.84 0.86 0.88 0.9 0.92 0.94φ

44.24.44.64.8

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pressure

contact number

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b)

FIG. 3: Results from new computer simulations. For all fitsφc =0.84005. (a) Average overlap per particle in units of the mean particleradius is linear inφ − φc. (b) P obtained from the Cauchy stresstensor (circles) and the force on the walls (squares) satisfy a powerlaw (φ − φc)

Ψ with ψ = 1.13; dashed line shows a linear law forcomparison. (c)Z (rattlers included) exhibits a power lawZ−Zc ∝(φ−φc)

β with Zc = 3.94 andβ = 0.5015.

boundaries. Since,φ = Ap/A, whereAp is the (presumablyfixed) area occupied by the disks,B = φ∂P/∂φ. Then,B ∝(φ−φc)

ψ−1, which gives a weak pressure variation ofB aboveφc. We note that anomalous results for the bulk modulus havebeen observed in acoustical experiments by Jia, and discussedby Makse et. al. [11], where the bulk modulus nearφc variedfaster withP than was previously expected because of changesin Z.

SinceP in Fig. 2 corresponds closely to expectations for alinear force law, we performed a computer simulation for apolydisperse system of 1950 particles with a linear force law(kn = 105N/m) without friction; details can be found in [12].In Fig. 3 the results are shown for a larger range in densitythan done in earlier studies. All the data in Fig. 3 can be fittedwith a single value for the transition density ofφc = 0.84005.While the average overlap per particle (equivalent of the de-formationδ for physical particles) is clearly linear inφ, thepressureP is not: P increases faster than linear with an ex-ponent close to the one found in the experiment.Z is alsoconsistent with a power-law exponent close to 0.5. With the

4

FIG. 4: Pressure vs. Z; Experimental data and a fit to the modelofHenkes and Chakraborty [7]. In this fit, the constantC defined in thetext is treated as an adjustable parameter. The other fittingparameteris Zc.

rattlers included,Z at φc, Zc = 3.94, is slightly below the iso-static value of 4 for a frictionless system of disks.

To connect with the predictions of Henkes and Chakraborty[7], we considerP− Pc vs. Z. The prediction from theirEq. (10) is equivalent to(P−Pc)/Pc = u− [(4u2 + 1)1/2 −

1]/2, where u = C(Z − Zc) and C = ε/αc is a system-dependent constant. Thus,ε is a measure of the grain elas-ticity, andε = 0 corresponds to completely rigid grains. Also,αc is the critical value forα. In fitting to this form, we mayadjustPc, (within reason)Zc, andC. In Fig. 4 we find rea-sonable although not perfect agreement with this prediction(aboveφc), and obtainZc = 3.04, which is close to the iso-static valueZc = 3.

We now turn to the rounding that we observe inZ quiteclose to the transition, and the background pressure that weobtain nearφc. One possible explanation is the friction be-tween the disks and the Plexiglas base. This could help freezein contact forces and contacts. However, a simple estimateof the upper bound for the friction with the base shows thatthis cannot be a significant effect, at least as regards the pres-sure background. To obtain an estimated upper bound forthe base friction onP, we assume that base friction can sup-port inter-grain contact forces corresponding to the maximumbase frictional force per grain,Ff = µbamg= 2.8×10−3 N,wherem is the mass of a grain andµba < 1 is the friction be-tween a particle and the base. AssumingZ inter-particle con-tacts and one particle-base contact per grain, we estimate theresulting upper bound on the perturbation to the pressure asδP ≃ (ZFf R)/(πR2) ≃ 0.22Z N/m, whereR is a disk radius.SinceZ ≃ 3, this pressure is almost two orders of magnitudetoo small to be of relevance. An additional issue concerns theanisotropy that is induced during compression or expansionby the apparatus. This induced anisotropy is difficult to avoidand/or relax close toφc even in the simulation, but it remains

small. It is visible in Fig. 1e, where a weak array of forcechains tends to slant from lower left to upper right. Amongother reasons, the anisotropy can be induced by wall frictiondue to the confining lateral boundaries of the biaxial appara-tus.

We conclude by noting that these experiments, the first ofwhich we are aware, demonstrate the critical nature of jam-ming in a real granular material. Our results take advantageofthe high accuracy in contact numberZ that is afforded whenthe particles are photoelastic.Z shows a very rapid rise at apacking densityφc = 0.8422. The fine resolution in densityallows us to see that the transition is not as sharply discontin-uous under the present experimental conditions as in the com-puter simulation. Aboveφc, Z and P follow power laws inφ−φc with respective exponentsβ of 0.5 to 0.6 andψ ≈ 1.1.The values for bothβ andψ are consistent with recent sim-ulation results forfrictionlessparticles. In addition, we findreasonable agreement with a mean field model of the granu-lar jamming transition, again for frictionless particles.Theseresults suggest that effects of friction on jamming are likelymodest, although perhaps not ignorable. That jamming in theexperiment occurs over a narrow, but finite range inφ seemsmostly to be caused by small residual shear stresses that areinduced by interactions with the walls confining the sample(not the base supporting the particles). The ability of a smallamount of shear to affect the jamming transition is interest-ing, and points to the need for a deeper understanding of theeffects of anisotropy.

This work was supported by NSF-DMR0137119, NSF-DMR0555431, NSF-DMS0244492, the US-Israel BinationalScience Foundation #2004391, and DFG-SP714/3-1. Wethank E. Aharonov, B. Chakraborty, D. J. Durian, M. vanHecke, C.S. O’Hern, and S. Torquato for helpful discussions.

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